On principal indecomposable degrees and Sylow subgroups

Abstract

We conjectured in Malle and Navarro (J Algebra 370:402–406, 2012) that a Sylow p-subgroup P of a finite group G is normal if and only if whenever p does not divide the multiplicity of $\chi \in {{\text {Irr}}}(G)$ in the permutation character $(1_P)^G$, then p does not divide the degree $\chi (1)$. In this note, we prove an analogue of this for p-Brauer characters.